Later in [1]}, we introduced a variant of Rauzy graph in which the vertex set is same as the Rauzy graph but any two vertices \((u,v)\)  form an arc iff \(u[\frac{|u|}{2}+1,|u|]=v[1,\frac{|v|}{2}]\)  i.e, half the length of the vertices are matched instead of \(|u|-1\)  to form an arc. The idea of sharing half the length of vertex was motivated by the encoding procedure of vertices and edges in to DNA strand, proposed by Adleman in [2]}. Some interesting structural properties of half range Rauzy graphs were studied in [1]}.
